Binomial Distribution

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BinomialDistribution

The binomial distribution gives the discrete probability distribution P_p(n|N) of obtaining exactly n successes out of N Bernoulli trials (where the result of each Bernoulli trial is true with probability p and false with probability q=1-p). The binomial distribution is therefore given by

P_p(n|N) = [画像:(N; n)p^nq^(N-n)]

(1)

= [画像:(N!)/(n!(N-n)!)p^n(1-p)^(N-n),]

(2)

where (N; n) is a binomial coefficient. The above plot shows the distribution of n successes out of N=20 trials with p=q=1/2.

The binomial distribution is implemented in the Wolfram Language as BinomialDistribution [n, p].

The probability of obtaining more successes than the n observed in a binomial distribution is

[画像: P=sum_(k=n+1)^N(N; k)p^k(1-p)^(N-k)=I_p(n+1,N-n), ]

(3)

where

[画像: I_x(a,b)=(B(x;a,b))/(B(a,b)), ]

(4)

B(a,b) is the beta function, and B(x;a,b) is the incomplete beta function.

The characteristic function for the binomial distribution is

phi(t)=(q+pe^(it))^N

(5)

(Papoulis 1984, p. 154). The moment-generating function M for the distribution is

M(t) = <e^(tn)>

(6)

= [画像:sum_(n=0)^(N)e^(tn)(N; n)p^nq^(N-n)]

(7)

= [画像:sum_(n=0)^(N)(N; n)(pe^t)^n(1-p)^(N-n)]

(8)

= [pe^t+(1-p)]^N

(9)

M^'(t) = N[pe^t+(1-p)]^(N-1)(pe^t)

(10)

M^('')(t) = N(N-1)[pe^t+(1-p)]^(N-2)(pe^t)^2+N[pe^t+(1-p)]^(N-1)(pe^t).

(11)

The mean is

mu = M^'(0)

(12)

= N(p+1-p)p

(13)

= Np.

(14)

The moments about 0 are

mu_1^' = mu=Np

(15)

mu_2^' = Np(1-p+Np)

(16)

mu_3^' = Np(1-3p+3Np+2p^2-3Np^2+N^2p^2)

(17)

mu_4^' = Np(1-7p+7Np+12p^2-18Np^2+6N^2p^2-6p^3+11Np^3-6N^2p^3+N^3p^3),

(18)

so the moments about the mean are

mu_2 = Np(1-p)=Npq

(19)

mu_3 = Np(1-p)(1-2p)

(20)

mu_4 = Np(1-p)[3p^2(2-N)+3p(N-2)+1].

(21)

The skewness and kurtosis excess are

gamma_1 = [画像:(1-2p)/(sqrt(Np(1-p)))]

(22)

= [画像:(q-p)/(sqrt(Npq))]

(23)

gamma_2 = [画像:(6p^2-6p+1)/(Np(1-p))]

(24)

= [画像:(1-6pq)/(Npq).]

(25)

The first cumulant is

kappa_1=np,

(26)

and subsequent cumulants are given by the recurrence relation

[画像: kappa_(r+1)=pq(dkappa_r)/(dp). ]

(27)

The mean deviation is given by

[画像: MD=sum_(k=0)^N|k-Np|(N; k)p^k(1-p)^(N-k). ]

(28)

For the special case p=q=1/2, this is equal to

MD = [画像:2^(-N)sum_(k=0)^(N)(N; k)|k-1/2N|]

(29)

= [画像:{(N!!)/(2(N-1)!!) for N odd; ((N-1)!!)/(2(N-2)!!) for N even,]

(30)

where N!! is a double factorial. For N=1, 2, ..., the first few values are therefore 1/2, 1/2, 3/4, 3/4, 15/16, 15/16, ... (OEIS A086116 and A086117). The general case is given by

[画像: MD=2(1-p)^(N-|_Np_|)p^(|_Np_|+1)(|_Np_|+1)(N; |_Np_|+1). ]

(31)

Steinhaus (1999, pp. 25-28) considers the expected number of squares S(n,N,s) containing a given number of grains n on board of size s after random distribution of N of grains,

S(n,N,s)=sP_(1/s)(n|N).

(32)

Taking N=s=64 gives the results summarized in the following table.

n S(n,64,64)

0 23.3591

1 23.7299

2 11.8650

3 3.89221

4 0.942162

5 0.179459

6 0.0280109

7 0.0036840

8 4.16639×10^(-4)

9 4.11495×10^(-5)

10 3.59242×10^(-6)

An approximation to the binomial distribution for large N can be obtained by expanding about the value n^~ where P(n) is a maximum, i.e., where dP/dn=0. Since the logarithm function is monotonic, we can instead choose to expand the logarithm. Let n=n^~+eta, then

ln[P(n)]=ln[P(n^~)]+B_1eta+1/2B_2eta^2+1/(3!)B_3eta^3+...,

(33)

where

[画像: B_k=[(d^kln[P(n)])/(dn^k)]_(n=n^~). ]

(34)

But we are expanding about the maximum, so, by definition,

[画像: B_1=[(dln[P(n)])/(dn)]_(n=n^~)=0. ]

(35)

This also means that B_2 is negative, so we can write B_2=-|B_2|. Now, taking the logarithm of (◇) gives

ln[P(n)]=lnN!-lnn!-ln(N-n)!+nlnp+(N-n)lnq.

(36)

For large n and N-n we can use Stirling's approximation

ln(n!) approx nlnn-n,

(37)

[画像:(d[ln(n!)])/(dn)] approx (lnn+1)-1

(38)

= lnn

(39)

[画像:(d[ln(N-n)!])/(dn)] approx [画像:d/(dn)[(N-n)ln(N-n)-(N-n)]]

(40)

= [画像:[-ln(N-n)+(N-n)(-1)/(N-n)+1]]

(41)

= -ln(N-n),

(42)

and

[画像: (dln[P(n)])/(dn) approx -lnn+ln(N-n)+lnp-lnq. ]

(43)

To find n^~, set this expression to 0 and solve for n,

[画像: ln((N-n^~)/(n^~)p/q)=0 ]

(44)

[画像: (N-n^~)/(n^~)p/q=1 ]

(45)

(N-n^~)p=n^~q

(46)

n^~(q+p)=n^~=Np,

(47)

since p+q=1. We can now find the terms in the expansion

B_2 = [画像:[(d^2ln[P(n)])/(dn^2)]_(n=n^~)]

(48)

= [画像:-1/(n^~)-1/(N-n^~)]

(49)

= [画像:-1/(Npq)]

(50)

= [画像:-1/(Np(1-p))]

(51)

B_3 = [画像:[(d^3ln[P(n)])/(dn^3)]_(n=n^~)]

(52)

= [画像:1/(n^~^2)-1/((N-n^~)^2)]

(53)

= [画像:(q^2-p^2)/(N^2p^2q^2)]

(54)

= [画像:(1-2p)/(N^2p^2(1-p)^2)]

(55)

B_4 = [画像:[(d^4ln[P(n)])/(dn^4)]_(n=n^~)]

(56)

= [画像:-2/(n^~^3)-2/((n-n^~)^3)]

(57)

= [画像:(2(p^2-pq+q^2))/(N^3p^3q^3)]

(58)

= [画像:(2(3p^2-3p+1))/(N^3p^3(1-p)^3)).]

(59)

BinomialGaussian

Now, treating the distribution as continuous,

[画像: lim_(N->infty)sum_(n=0)^NP(n) approx intP(n)dn=int_(-infty)^inftyP(n^~+eta)deta=1. ]

(60)

Since each term is of order 1/N∼1/sigma^2 smaller than the previous, we can ignore terms higher than B_2, so

P(n)=P(n^~)e^(-|B_2|eta^2/2).

(61)

The probability must be normalized, so

[画像: int_(-infty)^inftyP(n^~)e^(-|B_2|eta^2/2)deta=P(n^~)sqrt((2pi)/(|B_2|))=1, ]

(62)

and

P(n) = [画像:sqrt((|B_2|)/(2pi))e^(-|B_2|(n-n^~)^2/2)]

(63)

= [画像:1/(sqrt(2piNpq))exp[-((n-Np)^2)/(2Npq)].]

(64)

Defining sigma^2=Npq,

[画像: P(n)=1/(sigmasqrt(2pi))exp[-((n-n^~)^2)/(2sigma^2)], ]

(65)

which is a normal distribution. The binomial distribution is therefore approximated by a normal distribution for any fixed p (even if p is small) as N is taken to infinity.

If N->infty and p->0 in such a way that Np->lambda, then the binomial distribution converges to the Poisson distribution with mean lambda.

Let x and y be independent binomial random variables characterized by parameters n,p and m,p. The conditional probability of x given that x+y=k is

[画像: P(x=i|x+y=k)=(P(x=i,x+y=k))/(P(x+y=k)) =(P(x=i,y=k-i))/(P(x+y=k)) =(P(x=i)P(y=k-i))/(P(x+y=k)) =((n; i)p^i(1-p)^(n-i)(m; k-i)p^(k-i)(1-p)^(m-(k-i)))/((n+m; k)p^k(1-p)^(n+m-k)) =((n; i)(m; k-i))/((n+m; k)). ]

(66)

Note that this is a hypergeometric distribution.

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 531, 1987.Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 102-103, 1984.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Incomplete Beta Function, Student's Distribution, F-Distribution, Cumulative Binomial Distribution." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 219-223, 1992.Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 108-109, 1992.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.

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Binomial Distribution

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Weisstein, Eric W. "Binomial Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BinomialDistribution.html

Binomial Distribution

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